Theorem fvopab4ndm 7010

Description: Value of a function given by an ordered-pair class abstraction, outside of its domain. (Contributed by NM, 28-Mar-2008.)

Hypothesis

Ref	Expression
fvopab4ndm.1	⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}

Assertion

Ref	Expression
fvopab4ndm	⊢ (¬ 𝐵 ∈ 𝐴 → (𝐹‘𝐵) = ∅)

Distinct variable group:   𝑥,𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem fvopab4ndm

Step	Hyp	Ref	Expression
1		fvopab4ndm.1	. . . . 5 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
2	1	dmeqi 5885	. . . 4 ⊢ dom 𝐹 = dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
3		dmopabss 5899	. . . 4 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴
4	2, 3	eqsstri 3985	. . 3 ⊢ dom 𝐹 ⊆ 𝐴
5	4	sseli 3935	. 2 ⊢ (𝐵 ∈ dom 𝐹 → 𝐵 ∈ 𝐴)
6		ndmfv 6903	. 2 ⊢ (¬ 𝐵 ∈ dom 𝐹 → (𝐹‘𝐵) = ∅)
7	5, 6	nsyl5 160	1 ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐹‘𝐵) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∅c0 4288  {copab 5167  dom cdm 5652  ‘cfv 6525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-dm 5662  df-iota 6481  df-fv 6533

This theorem is referenced by:  fvmptndm  7011